On the semisimplicity of low-dimensional representations of semisimple groups in characteristic \(p\)
The main result of the paper is that any module of dimension at most \(\sqrt{p/3}\) for a semisimple algebraic group is semisimple if the characteristic of the ground field is \(p\). In the process of proving this result, the author also proves that the tensor product of two semisimple modules is semisimple provided the dimension of the tensor product module is at most \(p/f-h\) with \(f\) being the index of the root lattice in the weight lattice and \(h\) being the Coxeter number. The situation for finite groups of Lie type is also considered. It should be pointed out that Jantzen has proved that the modules with dimensions at most \(p\) are semisimple and \textit{J.-P. Serre} [Invent. Math. 116, 513-530 (1994; Zbl 0816.20014)] has proved a much stronger result for the tensor product of semisimple modules to be semisimple. The readers should be warned that the definition of alcoves in the paper and the alcoves later referred are not consistent.
scientific article; zbMATH DE number 763942